Potential Analysis

, Volume 37, Issue 2, pp 125–169 | Cite as

On a Class of Markov Semigroups on Discrete Ultra-Metric Spaces

  • Alexander Bendikov
  • Alexander Grigor’yanEmail author
  • Christophe Pittet


We consider a discrete ultra-metric space \(\left( X,d\right) \) with a measure m and construct in a natural way a symmetric Markov semigroup \( \left\{ P_{t}\right\} _{t\geq 0}\) in \(L^{2}\left( X,m\right) \) and the corresponding Markov process \(\left\{ \mathcal{X}_{t}\right\} \). We prove upper and lower bounds of its transition function and its Green function, give a criterion for the transience, and estimate its moments.


Markov chain Markov semigroup Markov generator Ultra-metric space Heat kernel Transition probability 

Mathematics Subject Classifications (2010)

Primary 60J05; Secondary 60J27 60J35 60B15 


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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Bendikov, A.: Asymptotic formulas for symmetric stable semigroups. Expo. Math. 12(4), 381–384 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bendikov, A., Bobikau, B., Pittet Ch.: Spectral properties of a class of random walks on locally finite groups (2011). arXiv:1102.1952v1
  4. 4.
    Bendikov, A., Pittet, Ch., Sauer, R.: Spectral distribution and l 2-isoperimetric profile of laplace operators on groups. Math. Ann. (2011, in press)Google Scholar
  5. 5.
    Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brofferio, S., Woess, W.: On transience of card shuffling. Proc. Am. Math. Soc. 129(5), 1513–1519 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Coulhon, T., Grigor’yan, A., Pittet, Ch.: A geometric approach to on-diagonal heat kernel low bounds on groups. Ann. Inst. Fourier (Grenoble) 51(6), 1763–1827 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Darling, D.A., Erdös, P.: On the recurrence of a certain chain. Proc. Am. Math. Soc. 13, 447–450 (1962)CrossRefGoogle Scholar
  9. 9.
    Fereig, N., Molchanov, S.A.: Random walks on abelian groups with an infinite number of generators. Vestn. Mosk. Univ. Ser. I Mat. Meh. 5, 22–29 (1978)MathSciNetGoogle Scholar
  10. 10.
    Flatto, L., Pitt, J.: Recurrence criteria for random walks on countable abelian groups. Ill. J. Math. 18, 1–19 (1974)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, vol. 47. AMS/IP Studies in Advanced Mathematics (2009)Google Scholar
  12. 12.
    Gromov. M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2 (Sussex, 1991). London Math. Soc. Lecture Notes Ser., vol. 182, pp. 1–295. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  13. 13.
    Kasymdzhanova, M.A.: Recurrence of invariant markov chains on a class of abelian groups. Vestn. Mosk. Univ. Ser. I Mat. Meh. 3, 3–7 (1981)MathSciNetGoogle Scholar
  14. 14.
    Kesten, H., Spitzer, F.: Random walks on countably infinite abelian groups. Acta Math. 114, 237–265 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lawler, G.F.: Recurrence and transience for a card shuffling model. Comb. Probab. Comput. 4(2), 133–142 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Del Muto, M., Figa-Talamanca, A.: Diffusion on locally compact ultrametric spaces. Expo. Math. 22, 197–211 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Del Muto, M., Figa-Talamanca, A.: Anisotropic diffusion on totally disconnected abelian groups. Pac. J. Math. 225, 221–229 (2006)zbMATHCrossRefGoogle Scholar
  18. 18.
    Pittet, Ch., Saloff-Coste, L.: Amenable groups, isoperimetric profiles and random walks. In: Geometric Group Theory (Canberra, 1996), pp. 293–316 (1999)Google Scholar
  19. 19.
    Pittet, Ch., Saloff-Coste, L.: On random walks on wreath products. Ann. Probab. 30(2), 948–977 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Saloff-Coste, L.: Opérateurs pseudo-différentiels sur certains groupes totalement discontinus. Stud. Math. (83), 205–228 (1986)Google Scholar
  21. 21.
    Saloff-Coste, L.: Probability on groups: random walks and invariant diffusions. Not. Am. Math. Soc. 48(9), 968–977 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Taibleson, M.H.: Fourier Analysis on Local Fields. Princeton Univ. Press (1975)Google Scholar
  23. 23.
    Varopoulos, N.Th. Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge Univ. Press, Cambridge (1992)Google Scholar
  24. 24.
    Weil, A.: Basic number theory. In: Classics in Mathematics. Springer, Berlin (1995) (Reprint of the 2nd edn. (1973))Google Scholar
  25. 25.
    Woess, W.: Random Walks on Infinite Graphs and Groups, vol. 138. Cambridge University Press (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alexander Bendikov
    • 1
  • Alexander Grigor’yan
    • 2
    Email author
  • Christophe Pittet
    • 3
  1. 1.Institute of MathematicsWrocław UniversityWrocławPoland
  2. 2.Department of MathematicsUniversity of BielefeldBielefeldGermany
  3. 3.CMIUniversité d’Aix-Marseille IMarseille Cedex 13France

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